Properties

Label 2-405-9.4-c3-0-7
Degree $2$
Conductor $405$
Sign $-0.173 - 0.984i$
Analytic cond. $23.8957$
Root an. cond. $4.88833$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.663 − 1.14i)2-s + (3.11 − 5.40i)4-s + (2.5 − 4.33i)5-s + (12.0 + 20.8i)7-s − 18.8·8-s − 6.63·10-s + (4.13 + 7.16i)11-s + (−43.5 + 75.4i)13-s + (15.9 − 27.6i)14-s + (−12.4 − 21.5i)16-s − 51.9·17-s − 88.5·19-s + (−15.5 − 27.0i)20-s + (5.49 − 9.50i)22-s + (−64.6 + 111. i)23-s + ⋯
L(s)  = 1  + (−0.234 − 0.406i)2-s + (0.389 − 0.675i)4-s + (0.223 − 0.387i)5-s + (0.650 + 1.12i)7-s − 0.834·8-s − 0.209·10-s + (0.113 + 0.196i)11-s + (−0.929 + 1.60i)13-s + (0.305 − 0.528i)14-s + (−0.194 − 0.336i)16-s − 0.740·17-s − 1.06·19-s + (−0.174 − 0.302i)20-s + (0.0532 − 0.0921i)22-s + (−0.585 + 1.01i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(23.8957\)
Root analytic conductor: \(4.88833\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :3/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6634893662\)
\(L(\frac12)\) \(\approx\) \(0.6634893662\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-2.5 + 4.33i)T \)
good2 \( 1 + (0.663 + 1.14i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (-12.0 - 20.8i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-4.13 - 7.16i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (43.5 - 75.4i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 51.9T + 4.91e3T^{2} \)
19 \( 1 + 88.5T + 6.85e3T^{2} \)
23 \( 1 + (64.6 - 111. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (135. + 234. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (112. - 194. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 70.5T + 5.06e4T^{2} \)
41 \( 1 + (-183. + 317. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-97.7 - 169. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (179. + 311. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 29.4T + 1.48e5T^{2} \)
59 \( 1 + (429. - 743. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-278. - 482. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-20.9 + 36.2i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 549.T + 3.57e5T^{2} \)
73 \( 1 + 185.T + 3.89e5T^{2} \)
79 \( 1 + (40.2 + 69.7i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (288. + 499. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 224.T + 7.04e5T^{2} \)
97 \( 1 + (-277. - 480. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.26854910877770581719742397096, −10.11295849969109473251308773306, −9.196265289292105783049383820408, −8.791642928380069270477681939818, −7.30452591676969450552428488825, −6.22627314069181475936230475075, −5.35260601465472335097228539087, −4.29262516568776356789584558125, −2.22382195018894353396034697457, −1.84965309016722036436336019522, 0.21093754209155160538193839883, 2.21028490583025403186974680587, 3.44224426983657987256786102788, 4.63721629702061447247842905307, 6.03088605813209659214780974707, 7.01544731130987774115162134401, 7.74322738759545770563292275176, 8.419813342176294774649231318788, 9.692936258148076612783853176309, 10.83301988423575403981985882841

Graph of the $Z$-function along the critical line